Maths help trig

What I did

I’m confused why they had to use that identity. I though squaring would make extra results —

Like I would have done cast and the other value would be 360-(theta)

You are right to be cautious about spurious solutions arising from squaring then rooting. However, in this case, since there are two possible values of @ corresponding to sin@ = 3/5, it is reasonable to assume that there may also be two possible values of cos@, and that those values will be given by the trig identity in question.
Another approach would be to draw a second triangle with an obtuse value of @ and proceed geometrically.